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arXiv:2011.14533 · Quantum Information and Computation

Unstructured search by random and quantum walk

Thomas G. Wong

Published Dec 28, 2021• 6 citations
arXivHTMLPDF

Abstract

The task of finding an entry in an unsorted list of $N$ elements famously takes $O(N)$ queries to an oracle for a classical computer and $O(\sqrt{N})$ queries for a quantum computer using Grover's algorithm. Reformulated as a spatial search problem, this corresponds to searching the complete graph, or all-to-all network, for a marked vertex by querying an oracle. In this tutorial, we derive how discrete- and continuous-time (classical) random walks and quantum walks solve this problem in a thorough and pedagogical manner, providing an accessible introduction to how random and quantum walks can be used to search spatial regions. Some of the results are already known, but many are new. For large $N$, the random walks converge to the same evolution, both taking $N \ln(1/\epsilon)$ time to reach a success probability of $1-\epsilon$. In contrast, the discrete-time quantum walk asymptotically takes $\pi\sqrt{N}/2\sqrt{2}$ timesteps to reach a success probability of $1/2$, while the continuous-time quantum walk takes $\pi\sqrt{N}/2$ time to reach a success probability of $1$.

DOI
10.26421/qic22.1-2-4

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The task of finding an entry in an unsorted list of NNN elements famously takes O(N)O(N)O(N) queries to an oracle for a classical computer and O(N)O(\sqrt{N})O(N​) queries for a quantum computer using Grover's algorithm. Reformulated as a spatial search problem, this corresponds to searching the complete graph, or all-to-all network, for a marked vertex by querying an oracle. In this tutorial, we derive how discrete- and continuous-time (classical) random walks and quantum walks solve this problem in a thorough and pedagogical manner, providing an accessible introduction to how random and quantum walks can be used to search spatial regions. Some of the results are already known, but many are new. For large NNN, the random walks converge to the same evolution, both taking Nln⁡(1/ϵ)N \ln(1/\epsilon)Nln(1/ϵ) time to reach a success probability of 1−ϵ1-\epsilon1−ϵ. In contrast, the discrete-time quantum walk asymptotically takes πN/22\pi\sqrt{N}/2\sqrt{2}πN​/22​ timesteps to reach a success probability of 1/21/21/2, while the continuous-time quantum walk takes πN/2\pi\sqrt{N}/2πN​/2 time to reach a success probability of 111.
Abstract The spatial search problem aims to find a marked vertex of a finite graph using a dynamic with two constraints: (1) the walker has no compass and (2) the walker can check whether a vertex is marked only after reaching it. This problem is a generalization of unsorted database search and has many applications to algorithms. Classical algorithms that solve the spatial search problem are based on random walks and the computational complexity is determined by the hitting time. On the other hand, quantum algorithms are based on quantum walks and the computational complexity is determined not only by the number of steps to reach a marked vertex, but also by the success probability, since we need to perform a measurement at the end of the algorithm to determine the walker’s position. In this work, we address the spatial search problem on Johnson graphs using the coined quantum walk model. Since Johnson graphs are vertex- and distance-transitive, we have found an invariant subspace of the Hilbert space, which aids in the calculation of the computational complexity. We have shown that, for every fixed diameter, the asymptotic success probability is 1/2 after taking π N / ( 2 2 ) steps, where N is the number of vertices of the Johnson graph.
The quantum-walk-based spatial search problem aims to find a marked vertex using a quantum walk on a graph with marked vertices. We describe a framework for determining the computational complexity of spatial search by continuous-time quantum walk on arbitrary graphs by providing a recipe for finding the optimal running time and the success probability of the algorithm. The quantum walk is driven by a Hamiltonian derived from the adjacency matrix of the graph modified by the presence of the marked vertices. The success of our framework depends on the knowledge of the eigenvalues and eigenvectors of the adjacency matrix. The spectrum of the Hamiltonian is subsequently obtained from the roots of the determinant of a real symmetric matrix M , the dimensions of which depend on the number of marked vertices. The eigenvectors are determined from a basis of the kernel of M . We show each step of the framework by solving the spatial searching problem on the Johnson graphs with a fixed diameter and with two marked vertices. Our calculations show that the optimal running time is O(N)O(\sqrt {N})O(N​) with an asymptotic probability of 1+ o (1), where N is the number of vertices.
Continuous-time quantum walks provide a natural framework to tackle the fundamental problem of finding a node among a set of marked nodes in a graph, known as spatial search. Whether spatial search by continuous-time quantum walk provides a quadratic advantage over classical random walks has been an outstanding problem. Thus far, this advantage is obtained only for specific graphs or when a single node of the underlying graph is marked. In this Letter, we provide a new continuous-time quantum walk search algorithm that completely resolves this: our algorithm can find a marked node in any graph with any number of marked nodes, in a time that is quadratically faster than classical random walks. The overall algorithm is quite simple, requiring time evolution of the quantum walk Hamiltonian followed by a projective measurement. A key component of our algorithm is a purely analogue procedure to perform operations on a state of the form e^{-tH^{2}}|ψ⟩, which, for a given Hamiltonian H, only requires evolving H for time scaling as sqrt[t]. This allows us to quadratically fast-forward the dynamics of a continuous-time classical random walk. The applications of our Letter thus go beyond the realm of quantum walks and can lead to new analog quantum algorithms for preparing ground states of Hamiltonians or solving optimization problems.
This paper examines the performance of spatial search where the Grover diffusion operator is replaced by continuous-time quantum walks on a class of interdependent networks. We prove that, for a set of optimal quantum walk times and marked vertex phase shifts, a deterministic algorithm for structured spatial search is established that finds the marked vertex with 100% probability. This improves on the Childs and Goldstone spatial search algorithm on the same class of graphs, which we show can only amplify to 50% probability. Our method uses $\ensuremath{\lceil}\frac{\ensuremath{\pi}}{2\sqrt{2}}\sqrt{N}\ensuremath{\rceil}$ marked vertex phase shifts for an NNN-vertex graph, making it comparable to Grover's algorithm for an unstructured search. It is expected that this framework can be readily extended to deterministic spatial search on other families of graph structures.
Quantum computing is a new field of computer science that leverage the quan-tum properties of atomic and subatomic particles for solving computational tasks.However, quantum computing provides very little intuition on how it works. Thismakes the identification of computational tasks with a potential quantum speedupand the elaboration of quantum algorithms for said tasks very difficult. In this context,this thesis focuses on a quantum dynamical system called Discrete Time QuantumWalk (DTQW). DTQWs are dynamical systems with discrete time and space whichare inspired from random walks. This thesis focuses on the problem of searchingan element in an unstructured database. It is known in the literature that searchingpan element with a DTQW on a grid has a quasi-square-root (i.e. O( n poly-log n))complexity in the dataset size n. This provides a significant quantum speedup overthe linear complexity of classical brute-force or random walk based algorithms. A firstcontribution of this thesis shows that, when two elements are searched, the worst casecomplexity becomes quasi-linear while the average case complexity remains quasi-square-root. Moving on from grids to arbitrary topologies, a second contributionintroduces a new model of DTQW onto the edges of an arbitrary graph. This modelunique features are: (i) the position which is on the edges; (ii) a polarity on each edgethat generalizes the notion of right and left we have on a line. A third contributionintroduces a model of quantum distributed computing and shows how this model onan arbitrary graph can reproduce the dynamic of a DTQW on the same graph.
Spatial search by continuous-time quantum walks on complex networks is focused on using a quantum walk in continuous time in order to find a single or multiple marked vertices within the complex network. The specific formalism used here is to consider a coupling constant that shifts the state of the quantum walker from the initial state to the target state, which is the marked vertex. The thesis begins with establishing the mathematical framework of network theory, quantum walks and numerical methods that will be used in the remainder of the thesis. Then spatial search by continuous-time quantum walk is studied on regular and semi-regular graphs, where most analytical results can be found. This will get us acquainted with spatial search by quantum walk. The complex networks studied are Barabasi-Albert graphs and the Internet network on the level of autonomous systems. Different renormalized and pruned versions of the Internet network are studied. The parameters of the quantum walk that are focused on are the optimal values for the coupling constant, success probability, time and search time.